Posts Tagged ‘mesh’

Inertia. It’s the force that keeps us in bed after the alarm has rung. It seems to have a life of its own, and today we’ll see how it comes into play in keeping other stationary objects at rest.

Last time we identified a specific point of contact between spur gear teeth in a gear train and introduced the opposing forces, F1 and F 2, generated there. Today we’ll see what these forces represent, identifying one of them as inertia.

So where do these forces come from? They’re forces generated by different means that converge at the same point of contact, the point at which gear teeth mesh. They follow a very specific geometric path to meet there, an imaginary straight line referred to as the line of action.

F1 is always generated by a source of mechanical energy. In our locomotive example introduced earlier in this blog series that source is an electric traction motor, upon which a driving gear is mounted. When the motor is energized, a driving force F1 is generated, which causes gear teeth on the driving gear to push against gear teeth of the driven gear.

Force F2 is not as straightforward to understand, because it’s not generated by a motor. Instead, it’s the resisting force that the weight of a stationary object poses against its being moved from an at-rest position, known as inertia. The heavier the object, the more inertia it presents with. Trains, of course, are extremely heavy, and to get them to move a great deal of inertia must be overcome. Inertia is also a factor in attempting to stop objects already in motion.

To get a stationary locomotive to move, mechanical energy must be transmitted from the driving gear that’s attached to its traction motor, then on to the driven gear attached to its axle. At their point of contact, the driving force of the motor, F1, is met by the resisting force of inertia, F2. In order for the train to move, F1 must be greater than F2. If F1 is less than or equal to F2, then the train won’t leave the station.

Next week we’ll animate our static image and watch the interplay between gear teeth, taking note of the line of action during their movement.

Last time we learned how spur gears mesh together to form a gear train and we examined a train consisting of just two gears, one being the driving gear, the other the driven gear. Today we’ll take a look at the geometry behind the smooth functioning of modern spur gear teeth when we identify their peculiar shape to be that of an involute curve.

The curved profile of spur gear teeth conforms to a type of mathematical curve found in geometry known as an involute. The involute profile of a spur gear tooth is shown in red below.

The mathematical notion of the involute was first presented in 1673 by Dutch mathematician Christiaan Huygens, in his book, Horologium Oscillatorium. Huygens’ book presents his studies on clock pendulums and the applied mathematics he used in an effort to predict their often erratic motion on ships at sea. His book ultimately dealt with far more than this, resulting in a treatise on the mathematical properties of the involutes of curves.

To see how an involute curve is formed, we’ll conduct a simple experiment. One end of string is attached with a tack to a circular object, like the yellow rod shown in the following illustration. The other end of string has a red ball attached to it.

Forming An Involute Curve

If we grab the ball and pull the string taught while wrapping the string around the rod, the ball’s path will form an involute curve due to the incremental shortening of the string that occurs as it wraps around the rod.

Next time we’ll see how the involuteprofile of gear teeth contributes to efficient mechanical energy transmission in gear trains.

Last time we covered the basic terminology of spur gears. Today we’ll see how they interact with one another to form a gear train, such as the one depicted below.

Meshing Spur Gears Form A Gear Train

A gear train is formed when the teeth of two or more gears mesh and work together for the purpose of powering a mechanical device.

A gear train can consist of as little as two gears, but trains can be so large as to contain dozens of gears, depending on the complexity of the device they are powering. But no matter how many gears are employed, there are certain key features that are shared by every gear train assembly. First, one gear within the train must be attached to a shaft rotated by a source of mechanical energy, such as an engine or electric motor. This gear is called the driving gear.

The second requirement of a gear train is that at least one gear other than the driving gear is mounted to the shaft of a rotating machine part. This gear is called the driven gear.

Locomotive Gear Train Consisting Of Two Gears

The illustration above shows an exploded view of a locomotive gear train assembly consisting of two gears. The driving gear is mounted to the shaft of an electric traction motor. The driven gear is mounted to the locomotive’s axle.

When a motor is attached to the axle, the two gears mesh together. The traction motor converts electrical energy into mechanical energy, which is supplied to the driving gear via the spinning motor’s shaft. The teeth of the driving gear then transmit the motor’s mechanical energy to the teeth of the driven gear, which then turn the locomotive’s wheels. It’s just one of countless operations that can be performed with gear train assemblies.

Next time we’ll examine the geometry behind modern spur gear tooth design.